Sequences and S...

$$14, 12, 21, 59, 231, 1149, ?$$

Find the next term of the series $$1728, 2744, 4096, 5832, 8000, 10648, ?$$

Find the missing number from the given alternatives.

If $$1,\ a_{1},\ a_{2},\ a_{3},......$$ and $$a_{8}$$ are nine, ninth roots of unity (taken in counter- clockwise sequence ) then $$|(2-a_{1})(2-a_{3})(2-a_{7})|=$$  

$$8, 31, 122, 485, 1936, 7739, ?$$

The sum of infinite series $$\begin{vmatrix} 1 & 2 \\ 6 & 4 \end{vmatrix}+\begin{vmatrix} \frac { 1 }{ 2 }  & 2 \\ 2 & 4 \end{vmatrix}+\begin{vmatrix} \frac { 1 }{ 4 }  & 2 \\ \frac { 2 }{ 3 }  & 4 \end{vmatrix}+.........$$

If $$\dfrac{\pi}{4}-1+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{11}-\dfrac{1}{13}+=0$$ then value of $$\dfrac{1}{1\times3}+\dfrac{1}{5\times7}+\dfrac{1}{9\times11}+\dfrac{1}{13\times15}+..$$ is

Let $$t_{r}=\frac{r}{1+r^{2}+r^{4}}$$ then, $$\lim_{n\rightarrow \infty }\sum_{r=1}^{n}t_{r}$$ equals

$$499, 622, 868, 1237, 1729, 2344, ?$$

the value of $$\sum\limits_{n = 1}^\infty  {\frac{{(3n + 1)}}{{n(4n + 4){4^n}}}} $$ is equal to